In quantum communication, two parties exchange information encoded in quantum states. Typically, the quantum states are specially defined properties of photons such as pairs of polarization states (e.g., 0° and 90°, or 45° and 135°) or circular basis states (e.g., left-handedness and right-handedness). Through the quantum communication (“QC”), the two parties produce a shared random series of bits known only to them, which can then be used as secret keys in subsequent encryption and decryption of messages.
A third party can, in theory, eavesdrop on the QC between the two parties. Such eavesdropping perturbs the QC, however, introducing anomalies that the two intended parties can detect. Using conventional communication, the two parties post-process the results of the QC to remove any partial information acquired by an eavesdropper, and form shared secret keys from the remaining information resulting from the QC.
For example, according to one general approach to QC, a transmitter sets the quantum state of binary information, makes a record of how it set the quantum state, and transmits the information. Table 1 shows an example of quantum states and bases for different polarizations of photons. For the bases and states shown in Table 1, the transmitter selects a basis (rectilinear or diagonal), sets the polarization state for a photon in the selected basis, and records the bit value (0 or 1), the selected sending basis and the time of transmission.
TABLE 1Example bases and quantum states.Basis01Rectilinear (+)90° 0°Diagonal (×)45°135° (or −45°)
A receiver receives the binary information, measures the quantum state of the information and makes a record of how it measured the quantum state. The measured state depends on how the receiver performs the measurement (e.g., with measuring basis of rectilinear or diagonal). The transmitter and receiver are expected to record different bit values in some instances because the transmitter and receiver at times set/measure the quantum-state-encoded information in different ways. Thus, after exchanging information in quantum states, the transmitter and receiver compare their records of how the quantum states were set and measured. For this comparison, the transmitter and receiver exchange information over a public channel. Then, the transmitter and receiver produce a shared series of bits (keys) from the encoded information for which quantum states were set and measured in the same way by the transmitter and receiver.
For the bases and states shown in Table 1, for example, the receiver selects a basis (rectilinear or diagonal), measures the polarization state in the selected basis, and records the measured bit value and measuring basis. No possible measuring basis can distinguish all four states, so the receiver essentially guesses either rectilinear or diagonal. If the measuring basis happens to match the sending basis, the receiver should measure the correct bit value. If the measuring basis does not match the sending basis, however, the measured bit value is as likely to be correct as incorrect. For example, if the sending basis is diagonal for the bit value 0 (polarization state of 45°) but the measuring basis is rectilinear, the measured bit values of 0 (90°) and 1 (0°) are equally likely. The transmitter and receiver compare the sending basis and measuring basis for a given photon, and keep the bit value for a photon if the sending basis and measuring basis match.
If an eavesdropper intercepts and measures a photon, the measurement perturbs the quantum state of the photon. The eavesdropper can only guess the original sending basis when it re-encodes and re-transmits the photon to the intended destination. At the time of measurement by the receiver, the eavesdropping is not detected. Instead, for subsets of the bit values for which sending basis and measuring basis are found to match, the transmitter and receiver compare parity values. The parity values should match exactly, if the system is appropriately tuned and free from imperfections in transmission and reception. Eavesdropping introduces noticeable discrepancies in the bit values, which allows the transmitter and receiver to detect the eavesdropping, correct the keys, and establish an upper limit on the eavesdropper's partial information.
An error-free bit string shared by the transmitter and receiver can then be privacy-amplified (e.g., by hashing with a hashing function) to reduce its length. (Or, bits can simply be dropped, but this lacks advantages of privacy amplification.) The final length of the shared bit string can depend on the number of errors detected. Shortening the shared bit string with privacy amplification reduces knowledge an eavesdropper might have to an arbitrarily low level—typically, much less than a single bit.
Unfortunately, practical QC systems such as those using optical systems that track remote receivers exhibit increased errors due to difficulties in establishing appropriate states of polarization (SOPs) at a receiver. Typically, a transmitted state of polarization from a fixed receiver varies or appears to vary at a receiver as the transmitter optical system is adjusted to track a moving receiver. For example, in earth to satellite communication, a ground based transmitter tracks satellite motion, thereby producing changes in SOP at the satellite receiver that are unrelated to the intended QC signal.